1. Chapter 2: Integers and Mathematical Induction
Discrete Mathematical Structures:
Theory and Applications

2. Learning Objectives Learn about the basic properties of integers
Become aware how integers are represented in computer memory
Explore how addition and subtraction operations are performed on binary numbers
Learn how the principle of mathematical induction is used to solve problems
Discrete Mathematical Structures: Theory and Applications

3. Learning Objectives
Learn about loop invariants and how they are used to prove the correctness of loops
Explore various properties of prime numbers
Learn about linear Diophantine equations and how to solve them
Discrete Mathematical Structures: Theory and Applications

4. Integers Properties of Integers
Discrete Mathematical Structures: Theory and Applications

5. Integers
Discrete Mathematical Structures: Theory and Applications

6. Integers
Discrete Mathematical Structures: Theory and Applications

7. Integers
Discrete Mathematical Structures: Theory and Applications

8. Integers
Discrete Mathematical Structures: Theory and Applications

9. Integers The div and mod operators div mod Examples: 8 div 5 = 1
a div b = the quotient of a and b obtained by dividing a on b.
Examples:
8 div 5 = 1
13 div 3 = 4
mod
a mod b = the remainder of a and b obtained by dividing a on b
8 mod 5 = 3
13 mod 3 = 1
Discrete Mathematical Structures: Theory and Applications

10. Integers
Discrete Mathematical Structures: Theory and Applications

11. Integers
Discrete Mathematical Structures: Theory and Applications

12. Integers
Discrete Mathematical Structures: Theory and Applications

13. Integers Relatively Prime Number
Discrete Mathematical Structures: Theory and Applications

14. Integers Least Common Multiples
Discrete Mathematical Structures: Theory and Applications

15. Representation of Integers in Computer
Electrical signals are used inside the computer to process information
Two types of signals
Analog
Continuous wave forms used to represent such things as sound
Examples: audio tapes, older television signals, etc.
Digital
Represent information with a sequence of 0s and 1s
Examples: compact discs, newer digital HDTV signals
Discrete Mathematical Structures: Theory and Applications

16. Representation of Integers in Computers
Digital Signals
0s and 1s – 0s represent low voltage, 1s high voltage
Digital signals are more reliable carriers of information than analog signals
Can be copied from one device to another with exact precision
Machine language is a sequence of 0s and 1s
The digit 0 or 1 is called a binary digit , or bit
A sequence of 0s and 1s is sometimes referred to as binary code
Discrete Mathematical Structures: Theory and Applications

17. Representation of Integers in Computers
Decimal System or Base-10
The digits that are used to represent numbers in base 10 are 0,1,2,3,4,5,6,7,8, and 9
Binary System or Base-2
Computer memory stores numbers in machine language, i.e., as a sequence of 0s and 1s
Octal System or Base-8
Digits that are used to represent numbers in base 8 are 0,1,2,3,4,5,6, and 7
Hexadecimal System or Base-16
Digits and letters that are used to represent numbers in base 16 are 0,1,2,3,4,5,6,7,8,9,A ,B ,C ,D ,E , and F
Discrete Mathematical Structures: Theory and Applications

18. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

19. Representation of Integers in Computers
Two’s Complements and Operations on Binary Numbers
In computer memory, integers are represented as binary numbers in fixed-length bit strings, such as 8, 16, 32 and 64
Assume that integers are represented as 8-bit fixed-length strings
Sign bit is the MSB (Most Significant Bit)
Leftmost bit (MSB) = 0, number is positive
Leftmost bit (MSB) = 1, number is negative
Discrete Mathematical Structures: Theory and Applications

20. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

21. Representation of Integers in Computers
One’s Complements and Operations on Binary Numbers
Discrete Mathematical Structures: Theory and Applications

22. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

23. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

24. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

25. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications

26. Representation of Integers in Computers
Discrete Mathematical Structures: Theory and Applications